What exactly does it mean to "mug somebody off" in British English?

I'm British and I do recognise the phrase, and you are correct it is usually used by Cockneys.

Here is a reference to the reality TV show The Only Way is Essex where someone feels that they have been made of fool (mug) of by his date, who is dancing with another man. She responds that she didn't make him look like a fool (she didn't mug him off).

In The Streets song 'Don't mug yourself' released in 2002, advice is given to a friend not to mug themselves over a girl (not to make a fool of themselves over a girl).

To be mugged off means that you are being made a fool of by someone taking advantage of you. There are two different meaning to the word mug that comes into context with you use this phrase.

• to mug someone, is take something from someone by use of force. So in this context taking someone of their respect/street cred.

• to be a mug, is to be fooled by someone, or to be taken advantage of.

It may not have any well-defined meaning as a phrase (I'm British, and I've never heard it).

"Mug" has a variety of uses, so a sentence like that is probably not an idiom at all. You should bear in mind that a film may go to some lengths to accentuate (or even make up) "cockney-isms" — don't mistake movie Cockney for British English, they are definitely not the same thing!

I'm from Essex and it's definitely a phrase we use all the time! It means to treat someone like an idiot, "to mug them off."

To be a 'mug' or to 'mug somebody off' basically means to take advantage of or to treat someone as an idiot/fool. This is 'street language' widely used across the UK in both youths and adults.